Probability. The probability of an event occurring is between 0 and 1 If an event is certain not to...
Transcript of Probability. The probability of an event occurring is between 0 and 1 If an event is certain not to...
Probability
Probability
• The probability of an event occurring is between 0 and 1
• If an event is certain not to happen, the probability is 0 eg: the probability of getting a 7 when you roll a die = 0
• If an event is sure to happen, the probability is 1 eg: the probability of getting either a head or a tail when you flip a coin = 1
• All other events have a probability between 0 and 1
Likely or unlikely?
Not likely LikelyImpossible to happen to happen Certain
Very unlikely Equal chance Very Likely to happen of happening to happen
Relative Frequency
This gives information about how often an event occurred compared with other events.
eg: Maths Exam results from 26 students
Exam % No. of students Relative Frequency
> 80 10
60 - 80 12
40 - 60 3
< 40 1
2610
= 0.38 (2 dp)
2612
= 0.46 (2dp)
263
= 0.12 (2 dp)
261
= 0.04 (2 dp)
Sample Space
The set of all possible outcomes is called the sample space.
eg. If a die is rolled the sample space is:{ 1, 2, 3, 4, 5, 6 }
eg. If a coin is flipped the sample space is:{ H, T }
eg. For a 2 child family the sample space is:{ BB, BG, GB, GG }
Equally likely outcomes
In many situations we can assume outcomes are equally likely.
When events are equally likely:
Equally likely outcomes may come from, for example: experiments with coins, dice, spinners and packs of cards
Probability = Number of favourable outcomes
Number of possible outcomes
Pr (getting a 5 when rolling a dice) =61
Favourable Outcomes are the results we wantPossible Outcomes are all the results that are possible - the SAMPLE SPACE
Examples:
Pr (even number on a dice) =21
63
Pr (J, Q, K or Ace in a pack of cards) =5216
= 134
Lattice DiagramsA spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. Graph the sample space and use it to give the probabilities:
(a) P(head and a 4) (b) P(head or a 4)
Sample spaceP(head and a 4) =
81
P(head or a 4) =
(those shaded)
85
Sample Space is {H1, H2, H3, H4, T1, T2, T3, T4}
A spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. We can work out the sample space by a lattice diagram or a tree diagram.
Lattice Diagram
Lattice DiagramsThe sample space when rolling 2 dice can be shown by the following lattice diagram:
1 2 3 4 5 6
1
2
3
4
5
6
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Pr (double) =366
=61
Pr (total ≥ 7) =3621
=127
Tree Diagrams for ProbabilityA tree diagram is a useful way to work out probabilities.
B
B
B
B
BB
B
G
G
G
G
G
G
G
eg: Show the possible combination of genders in a 3 child family
1st child
2nd child
3rd child Outcomes
BBB
BBGBGB
BGGGBB
GBGGGB
GGG
Pr (2 girls & a boy) =
83
Experimental Probability
When we estimate a probability based on an experiment, we call the probability by the term “relative frequency”.
Relative frequency =trialsofnumbertotal
outcomessuccessfulofnumber
The larger the number of trials, the closer the experimental probability (relative frequency) is to the theoretical probability.
Probability Relative Frequency
Theoretical Experimental
Relating factor “is the term we use in”
Pr (getting a 5 when rolling a dice) =61
Favourable Outcomes are the results we wantPossible Outcomes are all the results that are possible – (the sample space)
Examples:
Pr (even number on a dice) =21
63
Pr (J, Q, K or Ace in a pack of cards) =5216
=134
Complementary Events
When rolling a die, ‘getting a 5’ and ‘not getting a 5’ are complementary events. Their probabilities add up to 1.
Pr (getting a 5) =61 Pr (not getting a 5) =
6
5
Using Grids (Lattice Diagrams) to find Probabilities
red die
blue die621
1
2
3
4
5
6
543
●
● ●
●
●●
●●
●●●
●● ●
●
●
●
●
●
● ●
● ●●
●
●
●
●
●
●
●●
●
●
●
●
●
Pr (double) =366
=61
Pr (total ≥ 7) =3621
=127
coin
die654321
H
T ●
●
●
●
●
● ●
●
●
●
●
●
Rolling 2 dice:
Rolling a die & Flipping a coin:
Pr (tail and a 5) =12
1
Pr (tail or a 5) =12
7
Using Grids (Lattice Diagrams) to find Probabilities
1 2 3 4 5 6
1
2
3
4
5
6
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Pr (double) =366
=61
Pr (total ≥ 7) =3621
=127
Multiplying ProbabilitiesIn the previous lattice diagram, when rolling a die and flipping a coin,
coin
die654321
H
T ●
●
●
●
●
● ●
●
●
●
●
●Pr (tail) = Pr (5) =
6
1
2
1
Pr (tail and a 5) =2
1 x6
1=
12
1
So Pr (A and B) = Pr (A) x Pr (B)
example: Jo has probability ¾ of hitting a target, and Ann has probability of ⅓ of hitting a target. If they both fire simultaneously at the target, what is the probability that:
a) they both hit it b) they both miss itie Pr (Jo hits and Ann hits)= Pr (Jo hits) x Pr (Ann hits)= ¾ x ⅓
= ¼
ie Pr (Jo misses and Ann misses)ie Pr (Jo misses) x Pr (Ann misses)= ¼ x ⅔
= 6
1
Tree Diagrams to find ProbabilitiesIn the above example about Jo and Ann hitting targets, we canwork out the probabilities using a tree diagram.
Let H = hit, and M = miss
Jo’s results
H
M
¾
¼
Ann’s results
H
MH
M
⅓
⅔
⅓
⅔
OutcomesH and H
H and MM and H
M and M
Probability¾ x ⅓ = ¼
¾ x ⅔ = ½¼ x ⅓ = 12
1
¼ x ⅔ = 6
1
total = 1Pr (both hit) = ¼
Pr (both miss) = 6
1
Pr (only one hits) ie Pr (Jo or Ann hits) = ½ +12
1= 12
7
●
●or
ExpectationWhen flipping a coin the probability of getting a head is ½, therefore if we flip the coin 100 times we expect to get a head 50 times.
Expected Number = probability of an event occurring x the number of trials
eg: Each time a rugby player kicks for goal he has a ¾ chance of being successful. If, in a particular game, he has 12 kicks for goal, how many goals would you expect him to kick?
Solution: Pr (goal) = ¾
Number of trials = 12
Expected number = ¾ x 12 = 9 goals